Di;i-/.iii(KX or h:i\iH>.\i s.i.\iri.ti.s 95 



Soil HON ni nil. Til ii'iioMc !'k(iiii.i.m 



<)li\ ioiisly till- tcli-phoiu- prolilt-m is analogous to thi- |>n)l)l(.'ni of tin- 

 has containinj; an unknown ratio of white lialls. The corri-spondinj; 

 clt-nients in tlu- two proMt'ins ma>' l)e talnilated as follows: 



1st — l.O(M) lulls in l)ai{ versus .")(), 000 calls originated. 



■Jnd -UK) halls drawn versus 'MW rails ol)ser\e(l. 



;{rd -7 white halls drawn \-ersus !) calls delaved Tiiore than 10 

 seconds (i.e., defective with reference to .i particul.ir char- 

 acteristic). 

 4th — To the \W^ (Tossihle hypotheses with reference to the iniknown 

 per cent, of white halls correspond 40,000 possihlc h\-p()th- 

 eses with reference to the unknown per cent, of calls 

 delayed more than 10 seconds. 



'I'he prohlems differ in that a hall dr.iwii from the hat; is retnrnei! 

 l)cfore another drawinjj is made, whereas an ohservetl call is coni- 

 parahle to a h.dl heing drawn and not returned. With the iiumhiTs 

 in\oKe<l, however, the discrepancy may he ignored. 



A formula of the same form as (4) will, therefore, gi\e the answer 

 to our question. We may, however, suhstitute definite integrals 

 in place of the finite summations since the difference hetween any 

 two consecutive piossihle values for the unknown ratio is very small. 

 The integrals together with some desirahle transformations of them 

 will Ix- founfl in the apjx-ndi.x to this article. We will mention here, 

 however, that the transformations made inxoKe an arhitrary assump- 

 tion as to how the a priori existence prohahilit)' for the difTerent 

 hypotheses varies. As stated ahove in connection with Prof. Chrystal's 

 views, this is the phase of the subject which lends itself to consider- 

 ahlc difference of opinion. The reader who contemplates using the 

 curves emhodied in this article should read the ai>[)endix with s[)ecial 

 reference to the assumptions made. 



The attached cur\es Fig. 1 show graphically the conclusions to be 

 <lrawn from the mathematical anaKsis. A glance at the right hand 

 end of the curves will show that they are associated in pairs. The 

 upper curve of a pair slof)es df)wnward from left to right while its 

 mate slofx-s upward. 



Consider the pair of cur\es marked .Q'.i. Vor the abscissa 'M)0 

 they give as ordinates the values .0(i25 and .014. The interpretation 

 of these figures is as follows: if 300 observations gave 3 per cent, of 

 calls delayed then we may bet 



1st — 99 in 100 that the unknown percentage of calls (Ul.uid is )iot 

 greater than 6.25. 



